## Prime Factors

Number theory has fascinated mathematicians for years. Fundamental to number theory are numbers themselves, and the basic building blocks for numbers are prime numbers. A prime number is a counting number that only has two factors, itself and one. Counting numbers which have more than two factors (such as six, whose factors are 1, 2, 3 and 6), are said to be composite numbers. The number one only has one factor and is considered to be neither prime nor composite.

When a composite number is written as a product of all of its prime factors, we have the prime factorization of the number. For example, the number 72 can be written as a product of primes as: 72 =. The expression "" is said to be the prime factorization of 72. The Fundamental Theorem of Arithmetic states that every composite number can be factored uniquely (except for the order of the factors) into a product of prime factors. What this means is that how you choose to factor a number into prime factors makes no difference. When you are done, the prime factorizations are essentially the same. Examine the two factor trees for 72 given below.

When we get done factoring using either set of factors to start with, we still have three factors of two and two factors of three or . This would be true if we had started to factor 72 as 24 times 3, 4 times 18, or any other pair of factors for 72.

Knowing the rules for divisibility will be very helpful when seeking to write a number in prime factorization form. Since a number is divisible by two if it ends in either 0, 2, 4, 6, or 8, it should be noted that two is the only even prime number. Another way to factor a number other than using factor trees is to start dividing by prime numbers, as shown below.

Once again, we can see that 72 =. Another key idea in writing the prime factorization of a number is an understanding of exponents. An exponent tells how many times the base is used as a factor. In the prime factorization of 72 =, the two is used as a factor three times and the three is used as a factor twice.

When checking to see if a number is prime or not, you need only divide by those prime numbers which when squared remain less than the given number. For example to see if 131 is prime, you need only check for divisibility by 2, 3, 5, 7, and 11, since 132 = 169. If a prime number greater than 13 divided 131, then the other factor would have to be less than 13 and you would have checked those already.